There are twelve regular decagons in this polyhedron, and sixty irregular pentagons. If the pentagons were closer to regularity, this would qualify as a near-miss to the ninety-two Johnson Solids. It is not known how many of these “near-misses” exist — primarily because this group of polyhedra lacks a precise definition.
This polyhedron was discovered with the aid of Stella 4d, software you can try for yourself at http://www.software3d.com/stella.php.
A few days ago, I found a new near-miss to the 92 Johnson Solids. It appears on this blog, five posts ago, and looks a lot like what you see above — the differences are subtle, and will be explained below, after “near-miss” has been clarified.
A near-miss is a polyhedron which is almost a Johnson Solid. So what’s a Johnson Solid?
Well, consider all possible convex polyhedra which have only regular polygons as faces. Remove from this set the five Platonic Solids:
Next, remove the thirteen Archimedean Solids:
Now remove the infinite sets of prisms and antiprisms, the beginning of which are shown here:
What’s left? The answer to this question is known; it’s the set of Johnson Solids. It has been proven that there are exactly 92 of them:
When Norman Johnson systematically found all of these, and named them, in the late 1960s, he found a number of other polyhedra which were extremely close to being in this set. These are called the “near misses.” An example of a near-miss is the tetrated dodecahedron, which I co-discovered, and named, about a decade ago:
If you go to http://www.software3d.com/Stella.php, you can download a free trial version of software, Stella 4d, written by a friend of mine, Robert Webb (RW), which I used to generate this last image, as well as the rotating .gif which starts this post. (The still pictures were simply found using Google image-searches.) Stella 4d has a built-in library of near-misses, including the tetrated dodecahedron . . . but it doesn’t have all of them.
Well, why not? The reason is simple: the near-misses have no precise definition. They are simply “almost,” but not quite, Johnson Solids. In the case of the tetrated dodecahedron, what keeps it from being a Johnson Solid is the edges where yellow triangles meet other yellow triangles. These edges must be ~7% longer than the other edges, so the yellow triangles, unlike the other faces, are not quite regular — merely close.
There is no way to justify an arbitrary rule for just how close a near-miss must be to “Johnsonhood” be considered an “official” near-miss, so mathematicians have made no such rule. Research to find more near-misses is ongoing, and, due to the “fuzziness” of the definition, may never stop.
My informal test for a proposed near-miss is simple: I show it to RW, and if he thinks it’s close enough to include in the near-miss library in Stella 4d, then it passes. This new one did, but not until RW found a way to improve it, using something I don’t really understand called a “spring model.” What you see at the top of this post is the result. Unlike in the previous version, the green decagons here are regular, but at the expense of regularity in the (former) blue squares, now near-squarish trapezoids, as well as the yellow hexagons. The pink hexagons are slightly irregular in both versions, and the red pentagons are regular in both.
I’m eagerly anticipating the release of the next version of Stella 4d, for this near-miss will be in it. If I tell my students about this new discovery, they’ll want to know how much I got paid for it, which is, of course, nothing. I don’t know how to explain to them what it feels like to participate in the discovery of something — anything — which will survive me by a very long time. There’s nothing else quite like that feeling.
Now I just need to think of a good name for this thing!
[Update: the new version of Stella is now out, and this polyhedron is now included in it. As it turns out, I no longer need to think of a name for this polyhedron, for RW took care of that for me, naming it the “zonish truncated icosahedron” in Stella‘s built-in library of polyhedra.]
This is a face-based zonish truncated icosahedron.
I’ve only been looking for a new near-miss for a decade!
Software credit: see www.software3d.com/Stella.php.
The tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. I then independently rediscovered it in 2003, and named it, not learning of Doskey’s original discovery for several years after that.
It has 28 faces: twelve regular pentagons, arranged in four panels of three pentagons each; four equilateral triangles (shown in blue); and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This polyhedron has tetrahedral symmetry.
(All images here were produced using Stella 4d, which you may try for free, after downloading the trial version from this website: www.software3d.com/Stella.php.)
Consider all possible convex polyhedra which have regular polygons as faces. Remove from this set the five Platonic Solids:
Next, remove the thirteen Archimedean Solids:
Now remove the infinite sets of prisms and antiprisms, the beginning of which are shown here:
What’s left? The answer to this question is known; it’s the set of Johnson Solids. It has been proven that there are exactly 92 of them:
When Norman Johnson systematically found all of these, and named them, in the late 1960s, he found a number of other polyhedra which were extremely close to being in this set. These are called the “near misses.” An example of a near-miss is the tetrated dodecahedron:
If you go to http://www.software3d.com/Stella.php, you can download a free trial version of software, Stella 4d, written by a friend of mine, Robert Webb, which I used to generate this last image. This program has a built-in library of near-misses . . . but it doesn’t have all of them.
Well, why not? The reason is simple: the near-misses have no precise definition. They are simply “almost,” but not quite, Johnson Solids. In the case of the tetrated dodecahedron, what keeps it from being a Johnson Solid is the edges where yellow triangles meet other yellow triangles. These edges must be ~7% longer than the other edges, so the yellow triangles, unlike the other faces, are not quite regular — merely close.
There is no way to justify an arbitrary rule for just how close a near-miss must be to “Johnsonhood” be considered an “official” near-miss, so mathematicians have made no such rule. Research to find more near-misses is ongoing, and, due to the “fuzziness” of the definition, may never stop.
I’ve played a small part in such research, myself. I’ve also been asked how much I’ve been paid for doing this work, but that question misses the point. I’ve collected no money from this, and nobody gets involved in such research in order to get rich. Those of us who do such things are motivated by the desire to have fun through indulgence of mathematical curiosity. Our reward is the pure enjoyment of trying to figure things out, and, on really good days, actually doing so.
I’m having a good day. I’m looking at the Johnson Solids in a different way, purely for fun. I have found something that may be a blind alley, but, if my fellow geometricians show me that it is, that won’t erase the fun I have already had.
Here’s what I have found today. It is not a near-miss in the same way as the tetrated dodecahedron, but is related to the Johnson Solids in a different way. Other than a “heptadecahedron” (for its seventeen faces) it has no name, as of yet:
How is this different from traditional near-misses? Please examine the net (third image). In this heptadecahedron, all of these triangles, pentagons, and the one decagon are perfectly regular, unlike the situation with traditional near-misses. However, some faces, as you can see in the 3-d model, are made of multiple, coplanar equilateral triangles, joined together. In the blue faces, two such triangles form a rhombus; in the yellow faces, three such triangles form an isosceles trapezoid. Since they are coplanar and adjacent, they are one face each, not two, nor three. The dashed lines are not folded in the 3-d model, but merely show where the equilateral triangles are.
Traditional near-misses involve relaxation of the rules for Johnson Solids to permit polyhedra with not-quite-regular faces to join a new “club.”
Well, this heptadecahedron is in a different “club.” To join it, a polyhedron must fit the criteria for “Johnsonhood,” except that some faces may be formed by amalgamation of multiple, coplanar regular polygons.
My current subject of speculation is this: would this new club have an infinite or a finite number of members? If finite, it will, I think, be a larger number than 92. If finite, it will also be a more interesting topic to study.
I don’t know, yet, what answer this new problem has. I do know I am having fun, though. Also known: no one will pay me for this. No one needs to, either.
RobertLovesPi.net 